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yöneylem arastırasına giriş_tamsayılı programlama (integer programming)
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Operations Research
Solving IPs
There are two common approaches:
The technique based on dividing the problem into a number of
smaller problems in a tree search method called branch and
bound.
The method based on cutting planes (adding constraints to
force integrality).
Actually, all these approaches involve solving a series of LP
Categorization (w.r.t. Purpose)
General purpose methods will solve any IP but potentially
computationally ineffective (will only solve relatively small
problems);
Special purpose methods are designed for one particular
type of IP problem but potentially computationally more
effective.
Categorization (w.r.t. Algorithm) Optimal algorithms mathematically guarantee to find the
optimal solution
Heuristic algorithms are used to solve a problem by trial and error when an optimal algorithm approach is impractical. They hopefully find a good feasible solution that, in objective function terms, is close to the optimal solution.
Because the size of problem that we want to solve is beyond the computational limit of known optimal algorithms within the computer time we have available.
We could solve optimally but feel that this is not worth the effort (time, money, etc) we would expend in finding the optimal solution.
In fact it is often the case that a well-designed heuristic algorithm can give good quality (near-optimal) results.
Solution Algorithms Categories
General Purpose, Optimal
Branch and bound, cutting plane
General Purpose, Heuristic
Running a general purpose optimal algorithm and terminating
after a specified time
Special Purpose, Optimal
Tree search approaches based upon generating bounds via dual
ascent, lagrangean relaxation
Special Purpose, Heuristic
Bound based heuristics, tabu search, simulated annealing,
population heuristics (e.g. genetic algorithms), interchange
Solving IP
The Branch-and-Bound Method
B&B for Solving Pure IP Problems
B&B for Solving Mixed IP Problems
B&B for Solving Binary IP Problems
B&B for Solving Knapsack Problems
B&B for Solving Combinatorial Optimization Problems
Cutting Planes
B&B for Solving Pure IP Problems
The Telfa Corporation manufactures tables and chairs. A
table requires 1 hour of labor and 9 square board feet of
wood, and a chair requires 1 hour of labor and 5 square
board feet of wood. Currently, 6 hours of labor and 45
square board feet of wood are available. Each table
contributes $8 to profit, and each chair contributes $5 to
profit. Formulate and solve an IP to maximize Telfa’s
profit.
Solution
x1 number of tables manufactured
x2 number of chairs manufactured
max z = 8x1 + 5x2
s.t. x1 + x2 ≤ 6 (Labor constraint)
9x1 + 5x2 ≤ 45 (Wood constraint)
x1, x2 ≥ 0; x1, x2 integer
The branch-and-bound method begins by solving the LP
relaxation of the IP. If all the decision variables assume
integer values in the optimal solution to the LP relaxation,
then the optimal solution to the LP relaxation will be the
optimal solution to the IP. We call the LP relaxation
subproblem 1
the optimal solution to the LP relaxation is z=165/4, x1=
15/4, x2 = 9/4
(optimal z-value for IP) (optimal z-value for LP
relaxation)
Upper bound for Telfa’s profit= 165/4
Our next step is to partition the feasible region for the
LP relaxation in an attempt to find out more about the
location of the IP’s optimal solution. We arbitrarily choose
a variable that is fractional in the optimal solution to the
LP relaxation—say, x1.
Now observe that every point in the feasible region for
the IP must have either x1 ≤ 3 or x1 ≥ 4.
we “branch” on the variable x1 and create the following
two additional subproblems:
Subproblem 2 Subproblem 1 + Constraint x1 ≥ 4.
Subproblem 3 Subproblem 1 + Constraint x1 ≤ 3 .
We now choose any subproblem that has not yet been
solved as an LP.
We arbitrarily choose to solve subproblem 2.
the optimal solution to subproblem 2 is z = 41, x1 = 4, x2
= 9/5
The optimal solution to subproblem 2 did not yield an all-
integer solution, so we choose to use subproblem 2 to
create two new subproblems. Because x2 is the only
fractional variable in the optimal solution to subproblem
2, we branch on x2. We partition the feasible region for
subproblem 2 into those points having x2 ≥ 2 and x2 ≤ 1.
This creates the following two subproblems:
Subproblem 4 Subproblem 1+ Constraints x1 = 4 and
x2 ≥ 2 = subproblem 2 + Constraint x2 ≥ 2.
Subproblem 5 Subproblem 1 + Constraints x1= 4 and
x2 ≤ 1= subproblem 2 + Constraint x2 ≤ 1.
subproblem 4 is infeasible we place an X by subproblem
4.
the optimal solution to subproblem 5 is z = 365/9, x1 =
40/9, x2 =1.
we choose to partition subproblem 5’s
feasible region by branching on the fractional-valued
variable x1
Subproblem 6 Subproblem 5 + Constraint x1 ≥ 5.
Subproblem 7 Subproblem 5 + Constraint x1 ≤ 4.
B&B for Solving Mixed IP Problems
max z = 2x1 + x2
s.t. 5x1 + 2x2 ≤ 8
x1 + x2 ≤ 3
x1, x2 ≥ 0; x1 integer
The optimal solution of the
LP relaxation is z =11/3, x1 = 2/3, x2 = 7/3
x2 is allowed to be fractional, we do not branch on x2
Knapsack Problems
Any IP, that has only one constraint is referred to as a
knapsack problem.
B&B for Solving Knapsack Problems
max z = 8x1 + 11x2 + 6x3 + 4x4
s.t. 5x1 + 7x2 + 4x3 + 3x4 ≤ 14
xj = 0 or 1 (j = 1,2,3,4)
We compute the ratios:
r1 = 8 / 5 = 1.6
r2 = 11 / 7 = 1.57
r3 = 6 / 4 = 1.5
r4 = 4 / 3 = 1.33